Abstract
In this paper, we investigate the constrained assignment problem: a set of offers are to be assigned to a set of customers. There are constraints on both the number of available copies for each offer and the number of offers one customer can get. To measure the assignment gain, we have a score for each customer-offer pair, quantifying how beneficial it is for assigning a customer an offer. Additionally, a customer can get at most one copy of the same offer and at most one offer from the same category (an offer is associated with a category in a taxonomy, for example bakery, dairy, and so on). The objective is to optimize the assignment so that the sum of scores (global benefits) are maximized. We developed an auction algorithm for this problem and proved both its correctness and convergence. To show its effectiveness and efficiency, we compared it with heuristic algorithms and one minimum cost flow algorithm (network simplex). To show its scalability, we ran test cases of size up to 200 offers × 3,000,000 customers on a 16GB machine, where most of the linear/integer programming tools would fail under this setting. Finally, we transformed the auction algorithm into an equivalent belief propagation algorithm, and provided another convergence case for belief propagation on a loopy graph with node constraints.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andersen, D., Dahl, J., Vandenberghe, L.: CVXOPT: Python software for convex optimization, http://cvxopt.org/index.html
Bayati, M., Borgs, C., Chayes, J., Zecchina, R.: Belief propagation for weighted b-matchings on arbitrary graphs and its relation to linear programs with integer solutions. SIAM J. Discrete Math. 25, 989–1011 (2011)
Bayati, M., Shah, D., Sharma, M.: Max-product for maximum weight matching: convergence, correctness, and LP duality. IEEE Trans. Info. Theory 54, 1241–1251 (2008)
Bertsekas, D.P.: The auction algorithm for the transportation problem. Annals of Operations Research 20(1), 67–96 (1989)
Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods. Prentice-Hall (1989)
Blum, M., Floyd, B., Pratt, V., Rivest, R., Tarjan, B.: Linear time bounds for median computations. In: STOC, pp. 119–124 (1972)
Gamarnik, D., Shah, D., Wei, Y.: Belief propagation for min-cost network flow: Convergence and correctness. Operations Research 60, 410–428 (2012)
Kiraly, Z., Kovacs, P.: Efficient implementations of minimum-cost flow algorithms, http://arxiv.org/abs/1207.6381
Kiraly, Z., Kovacs, P.: LEMON graph library (COIN OR), http://lemon.cs.elte.hu/trac/lemon
Sanghavi, S.: Equivalence of LP relaxation and max-product for weighted matching in general graphs. In: IEEE Info. Theory Workshop, pp. 242–247 (2007)
Yuan, M., Jiang, C., Li, S., Shen, W., Pavlidis, Y., Li, J.: Message passing algorithm for the generalized assignment problem. In: Hsu, C.-H., Shi, X., Salapura, V. (eds.) NPC 2014. LNCS, vol. 8707, pp. 423–434. Springer, Heidelberg (2014)
Yuan, M., Li, S., Shen, W., Pavlidis, Y.: Belief propagation for minimax weight matching. Tech. rep., University of Illinois (2013)
Zavlanos, M.M., Spesivtsev, L., Pappas, G.J.: A distributed auction algorithm for the assignment problem. In: Proceedings of the 47th IEEE Conference on Decision and Control, pp. 1212–1217 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Yuan, M., Shen, W., Li, J., Pavlidis, Y., Li, S. (2015). Auction/Belief Propagation Algorithms for Constrained Assignment Problem. In: Ganguly, S., Krishnamurti, R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2015. Lecture Notes in Computer Science, vol 8959. Springer, Cham. https://doi.org/10.1007/978-3-319-14974-5_23
Download citation
DOI: https://doi.org/10.1007/978-3-319-14974-5_23
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-14973-8
Online ISBN: 978-3-319-14974-5
eBook Packages: Computer ScienceComputer Science (R0)